I am working through Guillemin & Pollack's Differential Topology, which shows that the Cartesian product of two manifolds $ X \times Y$ is itself a manifold. Moreover G&P shows that if $ \phi : W \to X $ is a parameterization of X, and $ \psi : U \to Y $ is a parameterization of Y, then we can define $ \phi \times \psi : W \times U \to X \times Y$ is a parameterization of $ X \times Y$, where $ \phi \times \psi (w,u) = (\phi (w), \psi (u)) $.
My question is about the derivative $ d (\phi \times \psi) _{(w,u)} $, intuitively, it would seems that the linear map defined by the derivative would be the direct sum of the Jacobian matrices of $d \phi _{w}$ and $d \psi _{u}$. This stems from the fact that $\phi$ has no dependence on the U coordinates, an likewise $\psi$ has no dependence on the W coordinates Is this correct?